Beware of commonly used approximations. Part I. Errors in forecasts
In the era of precision cosmology, establishing the correct magnitude of statistical errors in cosmological parameters is of crucial importance. However, widely used approximations in galaxy surveys analyses can lead to parameter uncertainties that are grossly mis-estimated, even in a regime where the theory is well understood (e.g., linear scales). These approximations can be introduced at three different levels: in the form of the likelihood, in the theoretical modelling of the observable and in the numerical computation of the observable. Their consequences are important both in data analysis through e.g., Markov Chain Monte Carlo parameter inference, and when survey instrument and strategy are designed and their constraining power on cosmological parameters is forecasted, for instance using Fisher matrix analyses. In this work, considering the galaxy angular power spectrum as the target observable, we report one example of approximation for each of such three categories: neglecting off-diagonal terms in the covariance matrix, neglecting cosmic magnification and using the Limber approximation on large scales. We show that these commonly used approximations affect the robustness of the analysis and lead, perhaps counter-intuitively, to unacceptably large mis-estimates of parameters errors (from few 10% up to few 100%) and correlations. Furthermore, these approximations might even spoil the benefits of the nascent multi-tracer and multi-messenger cosmology. Hence we recommend that the type of analysis presented here should be repeated for every approximation adopted in survey design or data analysis, to quantify how it may affect the results. To this aim, we have developed Multi CLASS, a new extension of CLASS that includes the angular power spectrum for multiple (galaxy and other tracers such as gravitational waves) populations. The public release of Multi CLASS is associated with this paper. ; Funding for this work was partially provided by the Spanish MINECO under projects AYA2014- 58747-P AEI/FEDER, UE, and MDM-2014-0369 of ICCUB (Unidad de Excelencia María de Maeztu). NB is supported by the Spanish MINECO under grant BES-2015-073372. JLB is supported by the Allan C. and Dorothy H. Davis Fellowship, and has been supported by the Spanish MINECO under grant BES-2015-071307, co-funded by the ESF during part of the development of this work. AR has received funding from the People Programme (Marie Curie Actions) of the European Union H2020 Programme under REA grant agreement number 706896 (COSMOFLAGS). GS was supported by the Erasmus+ for Trainership grant during the early stages of this work, subsequently by grant from the "Maria de Maeztu de Ci`encies del Cosmos" project mentioned above. GS is supported by the INFN INDARK PD51 grant. LV acknowledges support by European Union's Horizon 2020 research and innovation programme ERC (BePreSySe, grant agreement 725327).
